I'm not sure if there is a set prescription. The Suzuki method is quite popular as a way for beginners to advance. The way it all started was Suzuki began training children at younger ages than was common at the time. When these children began to be noticed everything pointed back to Suzuki as everyone wondered how such young children could be playing so well so young. Teachers from all around the world  came to hear the children and understand what Suzuki was doing. From there the curriculum spread and has become a staple training program for both children and adults.

The Suzuki books tend to use selected music as technique training in measured steps.Some of the material Suzuki wrote himself and some of it is the masters like Bach. Some of the exercises are technique designed to look like a song. The fingerings aren't always the ones you would use in a typical situation and are instead intended to be more difficult for learning purposes. 

I began to learn songs in Irish sessions. Later on I understood that this was leaving holes in my learning so now I augment with songs and mainly learn through the Suzuki method. Violin builds on a prior foundation. If you can't do X first well then you can't do Y. If you can't do Y well you can't do Z..........so in this respect attempting to circumvent or jump ahead to the more complex training won't work.

Don't ask me how I know that

Gordon Shumway said
What Andrew and Fiddlerman said, although warming up is maybe a very personal thing and varies depending on how far you have got in your studies. I'd recommend slow scales with much attention paid to tone quality (i.e. bowing) and especially to the intonation of each note (and the comfort of each left-hand finger). For me and Andrew there's orchestra to worry about, so after scales and as hard a study as I can cope with, I work on orchestra pieces.

There are two approaches to music -

a) make it fun (for kids or grown-up kids). This is why you'd play a tune a week, but maybe neglect good bowing.

b) make it good, what you (and I, a beginner 16 months ago at 58) are doing. For some people this approach kills their interest unless some fun can be introduced into it.

Suzuki isn't really a method unless you've got a Suzuki teacher. Without the teacher, Suzuki is just a collection of (sometimes rather tedious or tendentious) graded* tunes. Anyone else's graded collection is just as good. Here in the UK we tend to favour ABRSM (to be fair, also capable of being very boring). Suzuki seems to me to be a little cultic, so I am wary of it.

* such grading is not without contention. For example, in Suzuki 7 there's a Corelli piece that features much earlier in other collections, but maybe the bowing style affects its difficulty.

  

I admit I wasn't a big fan of Suzuki before I got a teacher. I jumped on board with the training after she recommended it. Agreed there is a huge difference between taking that method alone and taking it with a Suzuki teacher. The main difference I notice is I would have never repeated the material as often as she makes me play it if I were playing through the books all by myself. We continually go back to many of the things we had gone over before as re enforcement of a technique or for practice routines. We work on other material as well. 

I don't think there really is a good replacement for having an "in person" teacher if she or he is a good teacher. There are lesser replacements but none as good IMHO.

I find some of the Suzuki material fun to play and it feels good to learn a technique even if the process was grueling. I still play my Irish music every night plus other music.